Introduction: Total Probability = 1
Since \(|\psi|^2\) gives probability density, the total probability of finding the particle somewhere must be 1. This constrains the wavefunction's overall scale.
The Normalization Condition
\[\int_{-\infty}^{\infty} |\psi(x)|^2 dx = 1\]In Dirac notation: \(\langle\psi|\psi\rangle = 1\)
Normalizing a Wavefunction
If \(\psi\) is not normalized, find constant \(N\) such that \(N\psi\) is normalized:
\[N^2 \int |\psi|^2 dx = 1 \Rightarrow N = \left(\int |\psi|^2 dx\right)^{-1/2}\]Worked Examples
Example 1: Gaussian Normalization
Normalize \(\psi(x) = Ae^{-x^2/2a^2}\):
\[\int_{-\infty}^{\infty} |A|^2 e^{-x^2/a^2} dx = |A|^2 \sqrt{\pi}a = 1\] \[A = \left(\frac{1}{\pi a^2}\right)^{1/4}\]Example 2: Infinite Square Well
For \(\psi_n(x) = A\sin(n\pi x/L)\) on \([0, L]\):
\[\int_0^L |A|^2 \sin^2\left(\frac{n\pi x}{L}\right) dx = |A|^2 \frac{L}{2} = 1 \Rightarrow A = \sqrt{\frac{2}{L}}\]Example 3: Normalization Preserved in Time
Since \(\hat{U}(t)\) is unitary:
\[\langle\psi(t)|\psi(t)\rangle = \langle\psi(0)|\hat{U}^\dagger\hat{U}|\psi(0)\rangle = \langle\psi(0)|\psi(0)\rangle = 1\]The Quantum Connection
Normalization ensures that probabilities make sense. Square-integrable functions (those with finite \(\int|\psi|^2\)) form the physical Hilbert space \(L^2\). Non-normalizable functions like \(e^{ikx}\) (plane waves) require special treatment—they represent idealized states that are useful limits.